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is smooth formal. When it is (for example when a Q-∆ version of the ∂-∂ Lemma holds) there is a weak-Frobenius man-ifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, a necessary and sufficient condition for L to be smooth formal is presented. The condition is simply stated: it(More)
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability(More)
We describe a step toward quantizing deformation theory. The L∞ operad is encoded in a Hochschild cocyle • 1 in a simple universal algebra (P, • 0). This Hochschild cocyle can be extended naturally to a star product ⋆ = • 0+ • 1+ 2 • 2+ · · ·. The algebraic structure encoded in ⋆ is the properad Ω(coF rob) which, conjecturally, controls a quantization of(More)
Problem 1. Are the subspace and product topologies are consistent with each other? Let {X α } α∈A be a collection of topological spaces and let {Y α } be a collection of subsets; each Y α ⊂ X α. There are two ways to put a topology on Y = α∈A Y α : 1. Take the subspace topology on each Y α , then form the product topology on Y. 2. Take the product topology(More)